Two Estimators Remove Unit-Level Unobserved Heterogeneity

First Differences and LSDV

Airline Fares Depend on Unobserved Route and Time Characteristics

Two Sources of Omitted Variable Bias

\[ \begin{aligned} \log(\text{fare})_{it} &= \beta_0 + \beta_1\log(\text{distance})_i + \beta_2\text{competition}_{it} + e_{it} \\ e_{it} &= \gamma_i + \delta_t + u_{it} \end{aligned} \]

Controlling for Distance and Year Dummies Does Not Remove Route Heterogeneity

The Proposed Model Falls Short

Combining FD with Year Dummies Addresses Both Sources

Two Strategies for Two-Way Heterogeneity

(1) FD + year dummies, or (2) full LSDV with dummies for both units and time periods.

Industry-Wide Patent Growth Requires Flexible Time Effects

Year Dummies Capture Any Form of Common Time Trends

Year Dummy Coefficients Measure Growth Rates

Conditional Expectations Are The Tool to Interpret

\[\begin{align*} \mathbb{E}[\log(\text{patents})_{it} \mid t=2005] &= \beta_0 + \beta_1\log(\text{R\&D})_{it} \\ \mathbb{E}[\log(\text{patents})_{it} \mid t=2006] &= (\beta_0 + \beta_2) + \beta_1\log(\text{R\&D})_{it} \\ \mathbb{E}[\log(\text{patents})_{it} \mid t=2007] &= (\beta_0 + \beta_3) + \beta_1\log(\text{R\&D})_{it} \end{align*}\]

• Average log change in patents across all firms
• Geometric mean growth rate of patents in the industry, holding R&D constant
  • \(\beta_2\): 2005 to 2006
  • \(\beta_3\): 2005 to 2007

Year dummies measure growth rates between periods — not “the level in 2006 vs 2005.”

Interactions Allow the R&D Elasticity to Vary Over Time

Heterogeneous Elasticities

\[ \begin{align*} \log(\text{patents})_{it} &= \beta_0 + \beta_1\log(\text{R\&D})_{it} + \beta_2\mathbb{1}[t=2006] + \beta_3\mathbb{1}[t=2007] \\ &+ \beta_4(\log(\text{R\&D})_{it} \times \mathbb{1}[t=2006]) + \beta_5(\log(\text{R\&D})_{it} \times \mathbb{1}[t=2007]) \\ &+ e_{it} \end{align*} \]

Conditional means by year

\[\begin{align*} \mathbb{E}[\log(\text{patents})_{it} \mid t=2005] &= \beta_0 + \beta_1\log(\text{R\&D})_{it} \\ \mathbb{E}[\log(\text{patents})_{it} \mid t=2006] &= (\beta_0 + \beta_2) + (\beta_1 + \beta_4)\log(\text{R\&D})_{it} \\ \mathbb{E}[\log(\text{patents})_{it} \mid t=2007] &= (\beta_0 + \beta_3) + (\beta_1 + \beta_5)\log(\text{R\&D})_{it} \end{align*}\]

Level Shifts and Slope Shifts Are Separately Identified

Decomposing Differential Effects

No need to interpret \(\beta_4\) and \(\beta_5\) individually; the conditional means do the work.

Gender Wage Gaps Changed After the Mining Boom

Three Patterns from the Data

• Gender: Men earn a constant wage premium over women
• Time trend: Wages grow over time for both groups
• Structural break (2005): After the mining company arrives, the male premium widens

The Empirical Model Interacts Gender, Time, and Post-2005

Eight Coefficients for Four Groups

\[ \begin{align*} \log(\text{wages})_{it} &= \beta_0 + \beta_1\mathbb{1}[i\text{ is male}] + \beta_2 t + \beta_3\mathbb{1}[t \geq 2005] \\ &+ \beta_4(\mathbb{1}[i\text{ is male}] \times t) + \beta_5(\mathbb{1}[i\text{ is male}] \times \mathbb{1}[t \geq 2005]) \\ &+ \beta_6(t \times \mathbb{1}[t \geq 2005]) + \beta_7(t \times \mathbb{1}[t \geq 2005] \times \mathbb{1}[i\text{ is male}]) \\ &+ e_{it} \end{align*} \]

This model captures level differences, trends, and how both changed after 2005, separately for men and women.

Conditional Means: Pre-2005

Women and Men Before the Structural Break

Women before 2005 (base category)

\[ \mathbb{E}[\log(\text{wages})_{it} \mid \mathbb{1}[i \text{ is male}]=0,t<2005] = \beta_0 + \beta_2 t \]

Men before 2005

\[ \mathbb{E}[\log(\text{wages})_{it} \mid \mathbb{1}[i \text{ is male}]=1,t<2005] = (\beta_0 + \beta_1) + (\beta_2 + \beta_4)t \]

\(\beta_1\) shifts the intercept; \(\beta_4\) shifts the slope.

Conditional Means: Post-2005

Women and Men After the Structural Break

Women after 2005

\[ \mathbb{E}[\log(\text{wages})_{it} \mid \mathbb{1}[i \text{ is male}]=0,\; t \geq 2005] = (\beta_0 + \beta_3) + (\beta_2 + \beta_6)t\]

Men after 2005

\[\begin{align*} \mathbb{E}[\log(\text{wages})_{it} \mid \mathbb{1}[i \text{ is male}]=1,\; t \geq 2005] &= (\beta_0 + \beta_1 + \beta_3 + \beta_5) \\ &\quad + (\beta_2 + \beta_4 + \beta_6 + \beta_7)t \end{align*}\]

Each coefficient modifies either the intercept or slope for a specific group-period combination.

Taking Differences Isolates Each Coefficient’s Role

Condition on Group, Then Difference

Coefficient Signs Follow Directly from the Figure

Economic Interpretation

Panel Data Follows the Same Units Over Time

Definition and Structure

• \(y_{it}\), \(x_{it}\) for \(i = 1, \ldots, N\) and \(t = 1, \ldots, T\)
• Panel data: same units observed across multiple time periods
• Cross-section: one \(t\) only
• Repeated cross-section: same population, different individuals each period

Error decomposition

\[ e_{it} = \alpha_i + v_{it} \]

Panel Structure Enables Identification of Parameters

Identification vs Estimation

Example: cannot estimate returns to education via FD if education does not change over time.

Education Is Measured with Error in Both Periods

True vs Observed Variables

\[ \text{education}_{it} = \text{education}^{*}_{it} + e_{it} \]

Assumptions

• \(e_{it}\) uncorrelated with true education and other variables
• Education varies little over time for adults

Cross-Sectional Attenuation Bias Shrinks the Coefficient Towards Zero

The Baseline Problem

Population model

\[ \log(\text{wage})_i = \alpha + \beta\text{education}^{*}_i + \epsilon_i \]

Observed model substitutes \(\text{education}_i = \text{education}^{*}_i + e_i\)

Attenuation bias

\[ \text{plim}\;\hat\beta = \beta \cdot \frac{\text{Var}(\text{educ}^*)}{\text{Var}(\text{educ}^*) + \text{Var}(e)} \]

The ratio is less than 1, so the coefficient is biased towards zero (derived in Topic 6).

First-Differencing Increases the Measurement Error Variance

The Panel Data Paradox

FD of observed education

\[ \Delta\text{education}_i = \Delta\text{education}^{*}_{i} + (e_{i2} - e_{i1}) \]

If \(e_{i1}\) and \(e_{i2}\) are uncorrelated

\[ \text{Var}(e_{i2} - e_{i1}) = \text{Var}(e_{i1}) + \text{Var}(e_{i2}) \]

Roommate Nationality May Affect Student Grades

Self-Selection into Rooms Creates Endogeneity

Let \(\text{same}_{it} = \mathbb{1}[i\text{ has same-nationality roommate in } t]\)

\[ \text{grades}_{it} = \alpha + \beta\;\text{same}_{it} + e_{it} \]

• Exogeneity holds under random assignment of roommates
• If students choose roommates: an omitted equation determines room selection
• More outgoing students may prefer different nationalities and perform differently academically
• \(\text{Cov}(\text{same}_{it}, e_{it}) \neq 0\) arises from selection, not unobserved heterogeneity

Two Years of Data Require Roommate Changes

Panel Structure and Exogenous Mobility

Year 1

\(\text{grades}_{i1} = \alpha + \beta\;\text{same}_{i1} + \alpha_i + u_{i1}\)

Year 2

\(\text{grades}_{i2} = \alpha + \beta\;\text{same}_{i2} + \alpha_i + u_{i2}\)

First-differencing

\(\Delta\text{grades}_i = \beta\;\Delta\text{same}_i + \Delta u_i\)

• Critical: students must change roommates (\(\Delta\text{same}_i \neq 0\) for some)
• Exogenous mobility design — plausible if the university reassigns rooms

Random Assignment Plus Panel Data Strengthens Identification

You Get What You Pay For

• Random assignment of roommates:
  • \(\beta\) is identified even in cross-section
  • Panel adds precision: removes \(\alpha_i\) from the error \(\to\) smaller variance \(\to\) smaller standard errors
• Potential selection: panel data alone cannot solve the selection problem

Clustering Allows Arbitrary Within-Unit Correlation

A Conservative Fix for Panel Inference

Serial Correlation Only Affects Inference

The Three Pillars: Identification, Estimation, Inference

• Identification (AS1, AS3-AS5): Can we recover \(\beta\)? Not affected
• Estimation (AS2): How do we compute \(\hat{\beta}\)? Not affected
• Inference (AS2, AS7): Are our SEs, p-values, CIs valid?
  • Assumption 2 (i.i.d. sampling): same unit observed repeatedly → errors correlated within unit
  • Assumption 7 (homoskedasticity): error variance may differ across units
  • Both violations affect inference only — \(\hat{\beta}\) unchanged, SEs wrong

Measuring the Effect of Increased Force

All Municipalities Treated Simultaneously

A Linear Time Trend Restores Identification

Parametric but Estimable

\[\text{drug usage}_{it} = \mu\text{post}_t + \theta_i + \gamma t + e_{it}\]

When treatment varies only at the time level, time FE absorb it completely.

Proof

When We Do Not Know the Functional Form, Use Dummies

Nonparametric Estimation

\[ \log(\text{earnings})_{it} = \alpha_i + \theta_t + \sum_{j=17}^{85} \gamma_j\;\mathbb{1}[\text{age}_{it} = j] + e_{it} \]

• \(\alpha_i\): individual FE (absorbs ability, education, etc.)
• \(\theta_t\): time FE (absorbs aggregate macroeconomic trends)
• \(\gamma_j\): average difference in log-earnings between workers aged \(j\) and workers aged 16, holding constant \(\alpha_i\) and \(\theta_t\)
  • Nonparametric: no assumption on the shape of the age-earnings relation
  • Parametric: quadratic requires \(f(\text{age}) = \beta_1\;\text{age} + \beta_2\;\text{age}^2\)

Why log? APC problem

Imprecision at Extremes Reflects Thin Data

The Variance Formula for Dummy Variables

Each age dummy \(d_j = \mathbb{1}[\text{age}_{it} = j]\) is a binary variable with proportion \(p_j = n_j/n\):

\[ \text{Var}(\hat{\gamma}_j) \propto \frac{\sigma^2}{n \cdot p_j(1 - p_j)} \]

• Most workers aged 25-64 \(\Rightarrow\) \(p_j\) near its maximum \(\Rightarrow\) \(\text{Var}(\hat{\gamma}_j)\) small
• Few workers at ages 16-24 and 65-85 \(\Rightarrow\) \(p_j \approx 0\) \(\Rightarrow\) \(\text{Var}(\hat{\gamma}_j)\) large
• \(\hat{\gamma}_j\) at extreme ages has wide confidence intervals

Nonparametric flexibility comes at the cost of imprecision where data is thin.

Performance Pay Increases Firm-Level Productivity by 20%

But How Much Is Incentives vs Selection?

A firm introduces performance pay. The panel is unbalanced: some workers leave (exiters), some stay (stayers), some join (entrants).

\[ \log(\widehat{\text{productivity}})_{it} = \hat{\alpha}_i + \hat{\beta}\;\text{performance pay}_t \]

This Exercise Combines Three Previous Challenges

Collinearity (Q8) + Composition (Q10) + Time-Varying Controls (New)

\[ \text{productivity}_{it} = \beta_1\;\text{contingent}_t + \beta_2\;\text{weather}_t + \beta_3\;\text{width}_{it} + \beta_4\;\text{height}_{it} + e_{it} \]

Width and Height Are Time-AND-Individual Varying

Controls vs Fixed Effects Address Different Problems

\[\begin{align*} \text{productivity}_{it} &= \beta_1\;\text{contingent}_t + \beta_2\;\text{weather}_t + \beta_3\;\text{width}_{it} + \beta_4\;\text{height}_{it} \\ &+ \gamma_i + e_{it} \end{align*}\]

• Width and height vary across workers AND days — not captured by worker FE (\(\gamma_i\) absorbs time-invariant traits only)
• Still susceptible to Assumption 5: if correlated with \(e_{it}\), OVB persists
• Only workers in both periods contribute — churn shrinks effective sample

Controls handle time-varying confounders; FE handles time-invariant heterogeneity.

Call Centre Productivity Varies Across Operators, Leaders, and Days

Three Sources of Variation

Three-Way Fixed Effects Isolate Leader Quality

Rotation Is the Identification Condition

\[ \text{productivity}_{ijt} = \lambda_i + \mu_j + \theta_t + e_{ijt} \]

• \(\lambda_i\): operator FE — absorbs intrinsic worker ability
• \(\mu_j\): leader FE — the object of interest
• \(\theta_t\): day FE — absorbs common daily shocks
• Identification requires rotation: same operator observed under different leaders
  • Without rotation, \(\lambda_i\) and \(\mu_j\) are not separately identified

The F-Test Provides Evidence for Leader Quality Differences

Joint Significance of Leader Fixed Effects

Multiple fixed effects require sufficient rotation across dimensions for identification.

AKM (1999)

The Rubin Causal Model

Every Unit Has Two Potential Outcomes

From Individual Effects to Population Averages

The Average Treatment Effect

• We cannot compute \(\tau_{i}\) for any single person.
• But if all units come from the same DGP:
  • \(\text{ATE} = \mathbb{E}[y_{i}(1)] - \mathbb{E}[y_{i}(0)]\)
• Let \(T_{i} = 1\) if treated, \(0\) otherwise.
• What we actually observe:
  • \(\mathbb{E}[y_{i} \mid T_{i} = 1] - \mathbb{E}[y_{i} \mid T_{i} = 0]\)

When are these the same?

Only if who gets treated is unrelated to the potential outcomes.

The Observed Difference Decomposes into ATT and Selection Bias

The Add-and-Subtract Trick

Add and subtract \(\mathbb{E}[y(0) \mid T\!=\!1]\) inside the observed difference:

\[\begin{align*} \mathbb{E}[y \mid T\!=\!1] - \mathbb{E}[y \mid T\!=\!0] &= \mathbb{E}[y(1) \mid T\!=\!1] - \textcolor{orange}{\mathbb{E}[y(0) \mid T\!=\!1]} \\ &+ \textcolor{orange}{\mathbb{E}[y(0) \mid T\!=\!1]} - \mathbb{E}[y(0) \mid T\!=\!0] \\ &= \underbrace{\mathbb{E}[y(1) - y(0) \mid T\!=\!1]}_{\text{ATT}} \\ &+ \underbrace{\mathbb{E}[y(0) \mid T\!=\!1] - \mathbb{E}[y(0) \mid T\!=\!0]}_{\text{selection bias}} \end{align*}\]

DiD Recovers the Average Effect, Never the Individual

The Counterfactual Remains Unobserved

\[\text{ATT} = \mathbb{E}[y_{i}(1) - y_{i}(0) \mid T_{i} = 1]\]

• The counterfactual is still unobserved for every individual
• We use the control group’s trend as a proxy for the treated group’s missing trajectory
• Some individuals gain more, some less — we only recover the average

Some US States Simplified Their Tax Codes Between 1985 and 2005

Setting and Research Question

DiD Works Without Panel Data

Repeated Cross-Sections Suffice

\[\text{profit}_{it} = \alpha + \beta\;\mathbb{1}[\text{simple}_{i}] + \lambda\;\mathbb{1}[t = 2005] + \theta\;(\mathbb{1}[\text{simple}_{i}] \times \mathbb{1}[t = 2005]) + e_{it}\]

• \(\theta\) = the DiD estimator
  • Additional change in profit for simplified states beyond the common trend
• No panel data needed — DiD only requires group and time variation

The Identification Assumption

• Without tax simplification, both groups would have followed the same profit trajectory
• Violated if reforming states were already on a different growth path
• “As good as random” assignment strengthens this assumption

Police Encounters Vary by Officer-Citizen Race Combinations

Setting and Research Question

The Interaction Model Separates Race Effects from Match Effects

Cross-Sectional Interaction Model

\[\begin{align*} \text{force}_{ij} &= \beta_{0} + \beta_{1}\mathbb{1}[i\text{ is Black}] + \beta_{2}\mathbb{1}[j\text{ is Black}] \\ &+ \beta_{3}(\mathbb{1}[i\text{ is Black}] \times \mathbb{1}[j\text{ is Black}]) + e_{ij} \end{align*}\]

where \(i\) = citizen, \(j\) = officer

Coefficient Table

• \(\beta_{3}\) tests in-group/out-group bias
• NOT a DiD model — no time dimension

A Dutch City Banned Cannabis Sales to Foreign Students

Setting and Research Question

Individual Fixed Effects Absorb Level Differences

DiD with Panel Data

\[\text{grades}_{it} = \alpha_{i} + \lambda_{t} + \beta\;(\mathbb{1}[\text{non-DGB}_{i}] \times \mathbb{1}[\text{prohibited}_{t}]) + e_{it}\]

• \(\alpha_{i}\) absorbs the level difference between DGB and non-DGB students
• \(\beta\) = DiD estimator: the effect of cannabis prohibition on grades
• Recall Q8: universal treatment was collinear with time FE
  • Here, the control group (DGB students) breaks the collinearity

A School Meal Programme Was Assigned by Enrolment Number

Setting and Research Question

Random Assignment Makes Two Strategies Valid

Three Data Scenarios

Scenario 1 — Both years (DiD)

\[\begin{align*} \text{performance}_{it} &= \beta_{0} + \delta_{0}\mathbb{1}[t = 2013] + \beta_{1}\mathbb{1}[\text{even}_{i}] + \delta_{1}(\mathbb{1}[t = 2013] \times \mathbb{1}[\text{even}_{i}]) \\ &+ e_{it} \end{align*}\]

Scenario 2 — Only 2013 (simple differences)

\[\text{performance}_{i} = \eta_{0} + \eta_{1}\;\mathbb{1}[\text{even}_{i}] + e_{i}\]

• Valid because enrolment numbers are random — same logic as EE6.

Scenario 3 — Only 2012

• Impossible (programme did not exist)

The 2×2 Table Defines the DiD Estimator

Conditional Means by Group and Period

\[y_{it} = \beta_{0} + \delta_{0}\;\mathbb{1}[t = 2] + \beta_{1}\;\mathbb{1}[\text{Treatment}_{i}] + \delta_{1}\;(\mathbb{1}[t = 2] \times \mathbb{1}[\text{Treatment}_{i}]) + e_{it}\]

Two Paths to the Same Estimator

Algebraic Equivalence

Difference in changes

\[\delta_{1} = \underbrace{(\bar{y}_{T,\text{after}} - \bar{y}_{T,\text{before}})}_{\text{change for treated}} - \underbrace{(\bar{y}_{C,\text{after}} - \bar{y}_{C,\text{before}})}_{\text{change for control}}\]

Change in differences

\[\delta_{1} = \underbrace{(\bar{y}_{T,\text{after}} - \bar{y}_{C,\text{after}})}_{\text{gap after treatment}} - \underbrace{(\bar{y}_{T,\text{before}} - \bar{y}_{C,\text{before}})}_{\text{gap before treatment}}\]

The DiD Estimator Visually

Parallel Trends and the Treatment Effect

• Control (blue solid), Treated (gold solid), Counterfactual (gold dashed)
• Treatment effect = gap between actual treated and counterfactual at \(t = 2\)

A Retirement Reform Forced Some Officers to Work Longer

Setting and Research Question

Treatment Intensity Can Be Continuous

Heterogeneous Treatment in DiD

\[\text{sickness}_{it} = \beta\;(\Delta\text{years to work}_{i} \times \mathbb{1}[\text{post}]_{t}) + \text{controls}_{it} + \alpha_{i} + \lambda_{t} + e_{it}\]

• \(\Delta\text{years to work}_{i}\): extra years officer \(i\) must serve (continuous, not binary)
• \(\beta\): extra sickness days per additional year of forced service

First-Difference Derivation for General Panel Model

Algebra

Write the model for \(t = 1\) and \(t = 2\):

\[\begin{align*} y_{i1} &= \beta_0 + \beta_1 x_{i1,1} + \cdots + \beta_k x_{i1,k} + a_i + v_{i1} \\ y_{i2} &= (\beta_0 + \delta) + \beta_1 x_{i2,1} + \cdots + \beta_k x_{i2,k} + a_i + v_{i2} \end{align*}\]

Subtract: \(a_i - a_i = 0\).

\[ \Delta y_i = \delta + \beta_1\Delta x_{i1} + \beta_2\Delta x_{i2} + \cdots + \beta_k\Delta x_{ik} + \Delta v_i \]

First-Differencing Removes Time-Invariant Unobservables

• \(a_i\) eliminated by differencing: \(a_i - a_i = 0\)
• OLS on \(\Delta y_i\) is consistent under strict exogeneity: \(\mathbb{E}[\Delta v_i \mid \Delta x] = 0\)
• Time-invariant variables (e.g., distance, gender) also drop out
• Generalises to \(T > 2\): take consecutive differences for each pair \((t, t-1)\)

Return to Q1

Attenuation Factor: Cross-Section vs First-Differences

Cross-section

\[ \text{plim}\hat\beta_{\text{CS}} = \beta \cdot \frac{\text{Var}(\text{educ}^*)}{\text{Var}(\text{educ}^*) + \text{Var}(e)} \]

First-differences (assuming uncorrelated measurement errors)

\[ \text{plim}\;\hat\beta_{\text{FD}} = \beta \cdot \frac{\text{Var}(\Delta\text{educ}^*)}{\text{Var}(\Delta\text{educ}^*) + \text{Var}(e_{i1}) + \text{Var}(e_{i2})} \]

Why FD amplifies the bias

• Numerator shrinks: education barely changes → \(\text{Var}(\Delta\text{educ}^*) \ll \text{Var}(\text{educ}^*)\)
• Denominator doubles: \(\text{Var}(e_{i1}) + \text{Var}(e_{i2}) = 2\text{Var}(e)\)

Return to Q5

Why Log?

Three Reasons for Using Log-Earnings

1. Skewness: raw earnings are right-skewed
   • \(\log(\text{earnings})\) yields a more symmetric distribution, closer to normality (supporting AS7)
2. Multiplicative relationships: if earnings = base \(\times\) skill premium \(\times\) experience premium, then
   • \(\log(\text{earnings}) = \log(\text{base}) + \log(\text{skill premium}) + \log(\text{experience premium})\)
   • Log linearises multiplicative structures into additive ones
3. Growth rates: \(\Delta\log(\text{earnings}) \approx \%\Delta\text{earnings}\)
   • Differences in logs approximate percentage changes
   • The natural unit for comparing workers across different baseline earnings

Return to Q9

Age-Period-Cohort Identification Problem

Why Age, Time, and Individual FE Cannot All Be Included

OLS vs FE Decomposition — Formal

Stayers (\(S\)), Exiters (\(X\)), Entrants (\(N\))

Return to Q10

AKM: Matched Employer-Employee Framework

Abowd, Kramarz, and Margolis (1999, Econometrica)

The call centre exercise (Q12) builds on Abowd et al. (1999) and Fenizia (2022).

\[ \log(\text{wages})_{it} = \alpha_i + \psi_{J(i,t)} + x'_{it}\beta + e_{it} \]

• \(\alpha_i\): worker FE (ability, human capital)
• \(\psi_{J(i,t)}\): firm FE for the firm \(J\) employing worker \(i\) at time \(t\)
• Identification requires worker mobility across firms (exogenous mobility)
  • Same logic as Q12: operators rotate across teams
  • Without mobility, \(\alpha_i\) and \(\psi_{J(i,t)}\) not separately identified

Return to Q12

Why Day FE and Post Are Collinear

A Concrete Example with Four Days

Treatment starts on day 3:

\(\text{post}_t = d_3 + d_4\) — an exact linear combination of the day dummies. Stata would drop one variable automatically. The treatment effect \(\mu\) cannot be separated from the day effects.

Replacing Day FE with a Parametric Trend Restores Identification

• Resolution: replace \(T-1\) day dummies with a single parametric trend (\(\gamma\;\text{time}_t\))
  • Fewer parameters → no collinearity → \(\mu\) is identified
• Q8 application: all municipalities receive increased police force simultaneously — \(\text{post}_t\) is perfectly collinear with day FE
• Q11 application: contingent pay switches on a single date for all workers — same collinearity
• Trade-off: a linear trend is restrictive (assumes constant time effect) but estimable; day FE are flexible but break identification when treatment is universal

Return to Q8